On the partition dimension of circulant graph

Abstract

Let Λ = {B1,B2, . . . ,Bl} be an ordered l-partition of a connected graph G(V (G),E(G)). The partition representation of vertex x with respect to Λ is the l-vector, r(x|Λ) = (d(x,B1), d(x,B2), . . . , d(x,Bl)), where d(x,B) = min{d(x, y)|y ∈ B} is the distance between x and B. If the l - vectors r(x|Λ), for all x ∈ V (G) are distinct then l - partition is called a resolving partition. The least value of l for which there is a resolving l - partition is known as the partition dimension of G symbolized as pd(G).